When "Normalization Without Loss of Generality" Loses Generality

2026-03-29

AI Summary

This study investigates when normalization assumptions—commonly invoked in economics under the guise of “without loss of generality”—substantively affect model interpretation, counterfactual analysis, and statistical inference, thereby rendering identification results dependent on arbitrary modeling choices. Framing normalization as the selection of a representative from an equivalence class of observationally equivalent models, the paper establishes a rigorous criterion: counterfactual parameters are sensitive to normalization whenever they are non-constant over the equivalence class. The analysis reveals that normalization can artificially induce identification, exposing an extended trilemma at boundary singularities wherein fidelity, invariance, and regularity cannot be simultaneously satisfied. By integrating tools from differential geometry and econometric identification theory, the work demonstrates how normalization introduces coordinate singularities that distort the parameter space geometry, leading to substantive inferential biases in discrete choice, demand estimation, and network formation models.

Abstract

Normalization is ubiquitous in economics, and a growing literature shows that ``normalizations'' can matter for interpretation, counterfactual analysis, misspecification, and inference. This paper provides a general framework for these issues, based on the formalized notion of modeling equivalence that partitions the space of unknowns into equivalence classes, and defines normalization as a WLOG selection of one representative from each class. A counterfactual parameter is normalization-free if and only if it is constant on equivalence classes; otherwise any point identification is created by the normalization rather than by the model. Applications to discrete choice, demand estimation, and network formation illustrate the insights made explicit through this criterion. We then study two further sources of fragility: an extension trilemma establishes that fidelity, invariance, and regularity cannot simultaneously hold at a boundary singularity, while a normalization can itself introduce a coordinate singularity that distorts the topological and metric structures of the parameter space, with consequences for estimation and inference.

Problem

• normalization
• modeling equivalence
• counterfactual analysis
• parameter identification
• singularity

Innovation

• normalization
• modeling equivalence
• counterfactual invariance
• coordinate singularity
• extension trilemma